src/HOL/Tools/int_arith.ML
 author huffman Thu Mar 26 11:33:50 2009 -0700 (2009-03-26) changeset 30732 afca5be252d6 parent 30685 dd5fe091ff04 child 30802 f9e9e800d27e permissions -rw-r--r--
parameterize assoc_fold with is_numeral predicate
```     1 (* Authors: Larry Paulson and Tobias Nipkow
```
```     2
```
```     3 Simprocs and decision procedure for numerals and linear arithmetic.
```
```     4 *)
```
```     5
```
```     6 structure Int_Numeral_Simprocs =
```
```     7 struct
```
```     8
```
```     9 (*reorientation simprules using ==, for the following simproc*)
```
```    10 val meta_zero_reorient = @{thm zero_reorient} RS eq_reflection
```
```    11 val meta_one_reorient = @{thm one_reorient} RS eq_reflection
```
```    12 val meta_number_of_reorient = @{thm number_of_reorient} RS eq_reflection
```
```    13
```
```    14 (*reorientation simplification procedure: reorients (polymorphic)
```
```    15   0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a Int.*)
```
```    16 fun reorient_proc sg _ (_ \$ t \$ u) =
```
```    17   case u of
```
```    18       Const(@{const_name HOL.zero}, _) => NONE
```
```    19     | Const(@{const_name HOL.one}, _) => NONE
```
```    20     | Const(@{const_name Int.number_of}, _) \$ _ => NONE
```
```    21     | _ => SOME (case t of
```
```    22         Const(@{const_name HOL.zero}, _) => meta_zero_reorient
```
```    23       | Const(@{const_name HOL.one}, _) => meta_one_reorient
```
```    24       | Const(@{const_name Int.number_of}, _) \$ _ => meta_number_of_reorient)
```
```    25
```
```    26 val reorient_simproc =
```
```    27   Arith_Data.prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc);
```
```    28
```
```    29 (*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic isn't complicated by the abstract 0 and 1.*)
```
```    30 val numeral_syms = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym];
```
```    31
```
```    32
```
```    33 (** Utilities **)
```
```    34
```
```    35 fun mk_number T n = HOLogic.number_of_const T \$ HOLogic.mk_numeral n;
```
```    36
```
```    37 fun find_first_numeral past (t::terms) =
```
```    38         ((snd (HOLogic.dest_number t), rev past @ terms)
```
```    39          handle TERM _ => find_first_numeral (t::past) terms)
```
```    40   | find_first_numeral past [] = raise TERM("find_first_numeral", []);
```
```    41
```
```    42 val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
```
```    43
```
```    44 fun mk_minus t =
```
```    45   let val T = Term.fastype_of t
```
```    46   in Const (@{const_name HOL.uminus}, T --> T) \$ t end;
```
```    47
```
```    48 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
```
```    49 fun mk_sum T []        = mk_number T 0
```
```    50   | mk_sum T [t,u]     = mk_plus (t, u)
```
```    51   | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
```
```    52
```
```    53 (*this version ALWAYS includes a trailing zero*)
```
```    54 fun long_mk_sum T []        = mk_number T 0
```
```    55   | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
```
```    56
```
```    57 val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
```
```    58
```
```    59 (*decompose additions AND subtractions as a sum*)
```
```    60 fun dest_summing (pos, Const (@{const_name HOL.plus}, _) \$ t \$ u, ts) =
```
```    61         dest_summing (pos, t, dest_summing (pos, u, ts))
```
```    62   | dest_summing (pos, Const (@{const_name HOL.minus}, _) \$ t \$ u, ts) =
```
```    63         dest_summing (pos, t, dest_summing (not pos, u, ts))
```
```    64   | dest_summing (pos, t, ts) =
```
```    65         if pos then t::ts else mk_minus t :: ts;
```
```    66
```
```    67 fun dest_sum t = dest_summing (true, t, []);
```
```    68
```
```    69 val mk_diff = HOLogic.mk_binop @{const_name HOL.minus};
```
```    70 val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT;
```
```    71
```
```    72 val mk_times = HOLogic.mk_binop @{const_name HOL.times};
```
```    73
```
```    74 fun one_of T = Const(@{const_name HOL.one},T);
```
```    75
```
```    76 (* build product with trailing 1 rather than Numeral 1 in order to avoid the
```
```    77    unnecessary restriction to type class number_ring
```
```    78    which is not required for cancellation of common factors in divisions.
```
```    79 *)
```
```    80 fun mk_prod T =
```
```    81   let val one = one_of T
```
```    82   fun mk [] = one
```
```    83     | mk [t] = t
```
```    84     | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
```
```    85   in mk end;
```
```    86
```
```    87 (*This version ALWAYS includes a trailing one*)
```
```    88 fun long_mk_prod T []        = one_of T
```
```    89   | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
```
```    90
```
```    91 val dest_times = HOLogic.dest_bin @{const_name HOL.times} Term.dummyT;
```
```    92
```
```    93 fun dest_prod t =
```
```    94       let val (t,u) = dest_times t
```
```    95       in dest_prod t @ dest_prod u end
```
```    96       handle TERM _ => [t];
```
```    97
```
```    98 (*DON'T do the obvious simplifications; that would create special cases*)
```
```    99 fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
```
```   100
```
```   101 (*Express t as a product of (possibly) a numeral with other sorted terms*)
```
```   102 fun dest_coeff sign (Const (@{const_name HOL.uminus}, _) \$ t) = dest_coeff (~sign) t
```
```   103   | dest_coeff sign t =
```
```   104     let val ts = sort TermOrd.term_ord (dest_prod t)
```
```   105         val (n, ts') = find_first_numeral [] ts
```
```   106                           handle TERM _ => (1, ts)
```
```   107     in (sign*n, mk_prod (Term.fastype_of t) ts') end;
```
```   108
```
```   109 (*Find first coefficient-term THAT MATCHES u*)
```
```   110 fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
```
```   111   | find_first_coeff past u (t::terms) =
```
```   112         let val (n,u') = dest_coeff 1 t
```
```   113         in if u aconv u' then (n, rev past @ terms)
```
```   114                          else find_first_coeff (t::past) u terms
```
```   115         end
```
```   116         handle TERM _ => find_first_coeff (t::past) u terms;
```
```   117
```
```   118 (*Fractions as pairs of ints. Can't use Rat.rat because the representation
```
```   119   needs to preserve negative values in the denominator.*)
```
```   120 fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);
```
```   121
```
```   122 (*Don't reduce fractions; sums must be proved by rule add_frac_eq.
```
```   123   Fractions are reduced later by the cancel_numeral_factor simproc.*)
```
```   124 fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
```
```   125
```
```   126 val mk_divide = HOLogic.mk_binop @{const_name HOL.divide};
```
```   127
```
```   128 (*Build term (p / q) * t*)
```
```   129 fun mk_fcoeff ((p, q), t) =
```
```   130   let val T = Term.fastype_of t
```
```   131   in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
```
```   132
```
```   133 (*Express t as a product of a fraction with other sorted terms*)
```
```   134 fun dest_fcoeff sign (Const (@{const_name HOL.uminus}, _) \$ t) = dest_fcoeff (~sign) t
```
```   135   | dest_fcoeff sign (Const (@{const_name HOL.divide}, _) \$ t \$ u) =
```
```   136     let val (p, t') = dest_coeff sign t
```
```   137         val (q, u') = dest_coeff 1 u
```
```   138     in (mk_frac (p, q), mk_divide (t', u')) end
```
```   139   | dest_fcoeff sign t =
```
```   140     let val (p, t') = dest_coeff sign t
```
```   141         val T = Term.fastype_of t
```
```   142     in (mk_frac (p, 1), mk_divide (t', one_of T)) end;
```
```   143
```
```   144
```
```   145 (** New term ordering so that AC-rewriting brings numerals to the front **)
```
```   146
```
```   147 (*Order integers by absolute value and then by sign. The standard integer
```
```   148   ordering is not well-founded.*)
```
```   149 fun num_ord (i,j) =
```
```   150   (case int_ord (abs i, abs j) of
```
```   151     EQUAL => int_ord (Int.sign i, Int.sign j)
```
```   152   | ord => ord);
```
```   153
```
```   154 (*This resembles TermOrd.term_ord, but it puts binary numerals before other
```
```   155   non-atomic terms.*)
```
```   156 local open Term
```
```   157 in
```
```   158 fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
```
```   159       (case numterm_ord (t, u) of EQUAL => TermOrd.typ_ord (T, U) | ord => ord)
```
```   160   | numterm_ord
```
```   161      (Const(@{const_name Int.number_of}, _) \$ v, Const(@{const_name Int.number_of}, _) \$ w) =
```
```   162      num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
```
```   163   | numterm_ord (Const(@{const_name Int.number_of}, _) \$ _, _) = LESS
```
```   164   | numterm_ord (_, Const(@{const_name Int.number_of}, _) \$ _) = GREATER
```
```   165   | numterm_ord (t, u) =
```
```   166       (case int_ord (size_of_term t, size_of_term u) of
```
```   167         EQUAL =>
```
```   168           let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
```
```   169             (case TermOrd.hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
```
```   170           end
```
```   171       | ord => ord)
```
```   172 and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
```
```   173 end;
```
```   174
```
```   175 fun numtermless tu = (numterm_ord tu = LESS);
```
```   176
```
```   177 val num_ss = HOL_ss settermless numtermless;
```
```   178
```
```   179
```
```   180 (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
```
```   181 val add_0s =  thms "add_0s";
```
```   182 val mult_1s = thms "mult_1s" @ [thm"mult_1_left", thm"mult_1_right", thm"divide_1"];
```
```   183
```
```   184 (*Simplify inverse Numeral1, a/Numeral1*)
```
```   185 val inverse_1s = [@{thm inverse_numeral_1}];
```
```   186 val divide_1s = [@{thm divide_numeral_1}];
```
```   187
```
```   188 (*To perform binary arithmetic.  The "left" rewriting handles patterns
```
```   189   created by the Int_Numeral_Simprocs, such as 3 * (5 * x). *)
```
```   190 val simps = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym,
```
```   191                  @{thm add_number_of_left}, @{thm mult_number_of_left}] @
```
```   192                 @{thms arith_simps} @ @{thms rel_simps};
```
```   193
```
```   194 (*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
```
```   195   during re-arrangement*)
```
```   196 val non_add_simps =
```
```   197   subtract Thm.eq_thm [@{thm add_number_of_left}, @{thm number_of_add} RS sym] simps;
```
```   198
```
```   199 (*To evaluate binary negations of coefficients*)
```
```   200 val minus_simps = [@{thm numeral_m1_eq_minus_1} RS sym, @{thm number_of_minus} RS sym] @
```
```   201                    @{thms minus_bin_simps} @ @{thms pred_bin_simps};
```
```   202
```
```   203 (*To let us treat subtraction as addition*)
```
```   204 val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];
```
```   205
```
```   206 (*To let us treat division as multiplication*)
```
```   207 val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
```
```   208
```
```   209 (*push the unary minus down: - x * y = x * - y *)
```
```   210 val minus_mult_eq_1_to_2 =
```
```   211     [@{thm minus_mult_left} RS sym, @{thm minus_mult_right}] MRS trans |> standard;
```
```   212
```
```   213 (*to extract again any uncancelled minuses*)
```
```   214 val minus_from_mult_simps =
```
```   215     [@{thm minus_minus}, @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym];
```
```   216
```
```   217 (*combine unary minus with numeric literals, however nested within a product*)
```
```   218 val mult_minus_simps =
```
```   219     [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
```
```   220
```
```   221 structure CancelNumeralsCommon =
```
```   222   struct
```
```   223   val mk_sum            = mk_sum
```
```   224   val dest_sum          = dest_sum
```
```   225   val mk_coeff          = mk_coeff
```
```   226   val dest_coeff        = dest_coeff 1
```
```   227   val find_first_coeff  = find_first_coeff []
```
```   228   val trans_tac         = K Arith_Data.trans_tac
```
```   229
```
```   230   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
```
```   231     diff_simps @ minus_simps @ @{thms add_ac}
```
```   232   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
```
```   233   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
```
```   234   fun norm_tac ss =
```
```   235     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
```
```   236     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
```
```   237     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
```
```   238
```
```   239   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
```
```   240   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
```
```   241   val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
```
```   242   end;
```
```   243
```
```   244
```
```   245 structure EqCancelNumerals = CancelNumeralsFun
```
```   246  (open CancelNumeralsCommon
```
```   247   val prove_conv = Arith_Data.prove_conv
```
```   248   val mk_bal   = HOLogic.mk_eq
```
```   249   val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
```
```   250   val bal_add1 = @{thm eq_add_iff1} RS trans
```
```   251   val bal_add2 = @{thm eq_add_iff2} RS trans
```
```   252 );
```
```   253
```
```   254 structure LessCancelNumerals = CancelNumeralsFun
```
```   255  (open CancelNumeralsCommon
```
```   256   val prove_conv = Arith_Data.prove_conv
```
```   257   val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less}
```
```   258   val dest_bal = HOLogic.dest_bin @{const_name HOL.less} Term.dummyT
```
```   259   val bal_add1 = @{thm less_add_iff1} RS trans
```
```   260   val bal_add2 = @{thm less_add_iff2} RS trans
```
```   261 );
```
```   262
```
```   263 structure LeCancelNumerals = CancelNumeralsFun
```
```   264  (open CancelNumeralsCommon
```
```   265   val prove_conv = Arith_Data.prove_conv
```
```   266   val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less_eq}
```
```   267   val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} Term.dummyT
```
```   268   val bal_add1 = @{thm le_add_iff1} RS trans
```
```   269   val bal_add2 = @{thm le_add_iff2} RS trans
```
```   270 );
```
```   271
```
```   272 val cancel_numerals =
```
```   273   map Arith_Data.prep_simproc
```
```   274    [("inteq_cancel_numerals",
```
```   275      ["(l::'a::number_ring) + m = n",
```
```   276       "(l::'a::number_ring) = m + n",
```
```   277       "(l::'a::number_ring) - m = n",
```
```   278       "(l::'a::number_ring) = m - n",
```
```   279       "(l::'a::number_ring) * m = n",
```
```   280       "(l::'a::number_ring) = m * n"],
```
```   281      K EqCancelNumerals.proc),
```
```   282     ("intless_cancel_numerals",
```
```   283      ["(l::'a::{ordered_idom,number_ring}) + m < n",
```
```   284       "(l::'a::{ordered_idom,number_ring}) < m + n",
```
```   285       "(l::'a::{ordered_idom,number_ring}) - m < n",
```
```   286       "(l::'a::{ordered_idom,number_ring}) < m - n",
```
```   287       "(l::'a::{ordered_idom,number_ring}) * m < n",
```
```   288       "(l::'a::{ordered_idom,number_ring}) < m * n"],
```
```   289      K LessCancelNumerals.proc),
```
```   290     ("intle_cancel_numerals",
```
```   291      ["(l::'a::{ordered_idom,number_ring}) + m <= n",
```
```   292       "(l::'a::{ordered_idom,number_ring}) <= m + n",
```
```   293       "(l::'a::{ordered_idom,number_ring}) - m <= n",
```
```   294       "(l::'a::{ordered_idom,number_ring}) <= m - n",
```
```   295       "(l::'a::{ordered_idom,number_ring}) * m <= n",
```
```   296       "(l::'a::{ordered_idom,number_ring}) <= m * n"],
```
```   297      K LeCancelNumerals.proc)];
```
```   298
```
```   299
```
```   300 structure CombineNumeralsData =
```
```   301   struct
```
```   302   type coeff            = int
```
```   303   val iszero            = (fn x => x = 0)
```
```   304   val add               = op +
```
```   305   val mk_sum            = long_mk_sum    (*to work for e.g. 2*x + 3*x *)
```
```   306   val dest_sum          = dest_sum
```
```   307   val mk_coeff          = mk_coeff
```
```   308   val dest_coeff        = dest_coeff 1
```
```   309   val left_distrib      = @{thm combine_common_factor} RS trans
```
```   310   val prove_conv        = Arith_Data.prove_conv_nohyps
```
```   311   val trans_tac         = K Arith_Data.trans_tac
```
```   312
```
```   313   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
```
```   314     diff_simps @ minus_simps @ @{thms add_ac}
```
```   315   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
```
```   316   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
```
```   317   fun norm_tac ss =
```
```   318     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
```
```   319     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
```
```   320     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
```
```   321
```
```   322   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
```
```   323   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
```
```   324   val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
```
```   325   end;
```
```   326
```
```   327 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
```
```   328
```
```   329 (*Version for fields, where coefficients can be fractions*)
```
```   330 structure FieldCombineNumeralsData =
```
```   331   struct
```
```   332   type coeff            = int * int
```
```   333   val iszero            = (fn (p, q) => p = 0)
```
```   334   val add               = add_frac
```
```   335   val mk_sum            = long_mk_sum
```
```   336   val dest_sum          = dest_sum
```
```   337   val mk_coeff          = mk_fcoeff
```
```   338   val dest_coeff        = dest_fcoeff 1
```
```   339   val left_distrib      = @{thm combine_common_factor} RS trans
```
```   340   val prove_conv        = Arith_Data.prove_conv_nohyps
```
```   341   val trans_tac         = K Arith_Data.trans_tac
```
```   342
```
```   343   val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
```
```   344     inverse_1s @ divide_simps @ diff_simps @ minus_simps @ @{thms add_ac}
```
```   345   val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
```
```   346   val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
```
```   347   fun norm_tac ss =
```
```   348     ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
```
```   349     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
```
```   350     THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
```
```   351
```
```   352   val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}]
```
```   353   fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
```
```   354   val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
```
```   355   end;
```
```   356
```
```   357 structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
```
```   358
```
```   359 val combine_numerals =
```
```   360   Arith_Data.prep_simproc
```
```   361     ("int_combine_numerals",
```
```   362      ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"],
```
```   363      K CombineNumerals.proc);
```
```   364
```
```   365 val field_combine_numerals =
```
```   366   Arith_Data.prep_simproc
```
```   367     ("field_combine_numerals",
```
```   368      ["(i::'a::{number_ring,field,division_by_zero}) + j",
```
```   369       "(i::'a::{number_ring,field,division_by_zero}) - j"],
```
```   370      K FieldCombineNumerals.proc);
```
```   371
```
```   372 (** Constant folding for multiplication in semirings **)
```
```   373
```
```   374 (*We do not need folding for addition: combine_numerals does the same thing*)
```
```   375
```
```   376 structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
```
```   377 struct
```
```   378   val assoc_ss = HOL_ss addsimps @{thms mult_ac}
```
```   379   val eq_reflection = eq_reflection
```
```   380   fun is_numeral (Const(@{const_name Int.number_of}, _) \$ _) = true
```
```   381     | is_numeral _ = false;
```
```   382 end;
```
```   383
```
```   384 structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
```
```   385
```
```   386 val assoc_fold_simproc =
```
```   387   Arith_Data.prep_simproc
```
```   388    ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
```
```   389     K Semiring_Times_Assoc.proc);
```
```   390
```
```   391 end;
```
```   392
```
```   393 Addsimprocs [Int_Numeral_Simprocs.reorient_simproc];
```
```   394 Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
```
```   395 Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
```
```   396 Addsimprocs [Int_Numeral_Simprocs.field_combine_numerals];
```
```   397 Addsimprocs [Int_Numeral_Simprocs.assoc_fold_simproc];
```
```   398
```
```   399 (*examples:
```
```   400 print_depth 22;
```
```   401 set timing;
```
```   402 set trace_simp;
```
```   403 fun test s = (Goal s, by (Simp_tac 1));
```
```   404
```
```   405 test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
```
```   406
```
```   407 test "2*u = (u::int)";
```
```   408 test "(i + j + 12 + (k::int)) - 15 = y";
```
```   409 test "(i + j + 12 + (k::int)) - 5 = y";
```
```   410
```
```   411 test "y - b < (b::int)";
```
```   412 test "y - (3*b + c) < (b::int) - 2*c";
```
```   413
```
```   414 test "(2*x - (u*v) + y) - v*3*u = (w::int)";
```
```   415 test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
```
```   416 test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
```
```   417 test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
```
```   418
```
```   419 test "(i + j + 12 + (k::int)) = u + 15 + y";
```
```   420 test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
```
```   421
```
```   422 test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)";
```
```   423
```
```   424 test "a + -(b+c) + b = (d::int)";
```
```   425 test "a + -(b+c) - b = (d::int)";
```
```   426
```
```   427 (*negative numerals*)
```
```   428 test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
```
```   429 test "(i + j + -3 + (k::int)) < u + 5 + y";
```
```   430 test "(i + j + 3 + (k::int)) < u + -6 + y";
```
```   431 test "(i + j + -12 + (k::int)) - 15 = y";
```
```   432 test "(i + j + 12 + (k::int)) - -15 = y";
```
```   433 test "(i + j + -12 + (k::int)) - -15 = y";
```
```   434 *)
```
```   435
```
```   436 (*** decision procedure for linear arithmetic ***)
```
```   437
```
```   438 (*---------------------------------------------------------------------------*)
```
```   439 (* Linear arithmetic                                                         *)
```
```   440 (*---------------------------------------------------------------------------*)
```
```   441
```
```   442 (*
```
```   443 Instantiation of the generic linear arithmetic package for int.
```
```   444 *)
```
```   445
```
```   446 structure Int_Arith =
```
```   447 struct
```
```   448
```
```   449 (* Update parameters of arithmetic prover *)
```
```   450
```
```   451 (* reduce contradictory =/</<= to False *)
```
```   452
```
```   453 (* Evaluation of terms of the form "m R n" where R is one of "=", "<=" or "<",
```
```   454    and m and n are ground terms over rings (roughly speaking).
```
```   455    That is, m and n consist only of 1s combined with "+", "-" and "*".
```
```   456 *)
```
```   457
```
```   458 val zeroth = (symmetric o mk_meta_eq) @{thm of_int_0};
```
```   459
```
```   460 val lhss0 = [@{cpat "0::?'a::ring"}];
```
```   461
```
```   462 fun proc0 phi ss ct =
```
```   463   let val T = ctyp_of_term ct
```
```   464   in if typ_of T = @{typ int} then NONE else
```
```   465      SOME (instantiate' [SOME T] [] zeroth)
```
```   466   end;
```
```   467
```
```   468 val zero_to_of_int_zero_simproc =
```
```   469   make_simproc {lhss = lhss0, name = "zero_to_of_int_zero_simproc",
```
```   470   proc = proc0, identifier = []};
```
```   471
```
```   472 val oneth = (symmetric o mk_meta_eq) @{thm of_int_1};
```
```   473
```
```   474 val lhss1 = [@{cpat "1::?'a::ring_1"}];
```
```   475
```
```   476 fun proc1 phi ss ct =
```
```   477   let val T = ctyp_of_term ct
```
```   478   in if typ_of T = @{typ int} then NONE else
```
```   479      SOME (instantiate' [SOME T] [] oneth)
```
```   480   end;
```
```   481
```
```   482 val one_to_of_int_one_simproc =
```
```   483   make_simproc {lhss = lhss1, name = "one_to_of_int_one_simproc",
```
```   484   proc = proc1, identifier = []};
```
```   485
```
```   486 val allowed_consts =
```
```   487   [@{const_name "op ="}, @{const_name "HOL.times"}, @{const_name "HOL.uminus"},
```
```   488    @{const_name "HOL.minus"}, @{const_name "HOL.plus"},
```
```   489    @{const_name "HOL.zero"}, @{const_name "HOL.one"}, @{const_name "HOL.less"},
```
```   490    @{const_name "HOL.less_eq"}];
```
```   491
```
```   492 fun check t = case t of
```
```   493    Const(s,t) => if s = @{const_name "HOL.one"} then not (t = @{typ int})
```
```   494                 else s mem_string allowed_consts
```
```   495  | a\$b => check a andalso check b
```
```   496  | _ => false;
```
```   497
```
```   498 val conv =
```
```   499   Simplifier.rewrite
```
```   500    (HOL_basic_ss addsimps
```
```   501      ((map (fn th => th RS sym) [@{thm of_int_add}, @{thm of_int_mult},
```
```   502              @{thm of_int_diff},  @{thm of_int_minus}])@
```
```   503       [@{thm of_int_less_iff}, @{thm of_int_le_iff}, @{thm of_int_eq_iff}])
```
```   504      addsimprocs [zero_to_of_int_zero_simproc,one_to_of_int_one_simproc]);
```
```   505
```
```   506 fun sproc phi ss ct = if check (term_of ct) then SOME (conv ct) else NONE
```
```   507
```
```   508 val lhss' =
```
```   509   [@{cpat "(?x::?'a::ring_char_0) = (?y::?'a)"},
```
```   510    @{cpat "(?x::?'a::ordered_idom) < (?y::?'a)"},
```
```   511    @{cpat "(?x::?'a::ordered_idom) <= (?y::?'a)"}]
```
```   512
```
```   513 val zero_one_idom_simproc =
```
```   514   make_simproc {lhss = lhss' , name = "zero_one_idom_simproc",
```
```   515   proc = sproc, identifier = []}
```
```   516
```
```   517 val add_rules =
```
```   518     simp_thms @ @{thms arith_simps} @ @{thms rel_simps} @ @{thms arith_special} @
```
```   519     [@{thm neg_le_iff_le}, @{thm numeral_0_eq_0}, @{thm numeral_1_eq_1},
```
```   520      @{thm minus_zero}, @{thm diff_minus}, @{thm left_minus}, @{thm right_minus},
```
```   521      @{thm mult_zero_left}, @{thm mult_zero_right}, @{thm mult_Bit1}, @{thm mult_1_right},
```
```   522      @{thm minus_mult_left} RS sym, @{thm minus_mult_right} RS sym,
```
```   523      @{thm minus_add_distrib}, @{thm minus_minus}, @{thm mult_assoc},
```
```   524      @{thm of_nat_0}, @{thm of_nat_1}, @{thm of_nat_Suc}, @{thm of_nat_add},
```
```   525      @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1}, @{thm of_int_add},
```
```   526      @{thm of_int_mult}]
```
```   527
```
```   528 val nat_inj_thms = [@{thm zle_int} RS iffD2, @{thm int_int_eq} RS iffD2]
```
```   529
```
```   530 val int_numeral_base_simprocs = Int_Numeral_Simprocs.assoc_fold_simproc :: zero_one_idom_simproc
```
```   531   :: Int_Numeral_Simprocs.combine_numerals
```
```   532   :: Int_Numeral_Simprocs.cancel_numerals;
```
```   533
```
```   534 val setup =
```
```   535   Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
```
```   536    {add_mono_thms = add_mono_thms,
```
```   537     mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} :: mult_mono_thms,
```
```   538     inj_thms = nat_inj_thms @ inj_thms,
```
```   539     lessD = lessD @ [@{thm zless_imp_add1_zle}],
```
```   540     neqE = neqE,
```
```   541     simpset = simpset addsimps add_rules
```
```   542                       addsimprocs int_numeral_base_simprocs
```
```   543                       addcongs [if_weak_cong]}) #>
```
```   544   arith_inj_const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) #>
```
```   545   arith_discrete @{type_name Int.int}
```
```   546
```
```   547 val fast_int_arith_simproc =
```
```   548   Simplifier.simproc (the_context ())
```
```   549   "fast_int_arith"
```
```   550      ["(m::'a::{ordered_idom,number_ring}) < n",
```
```   551       "(m::'a::{ordered_idom,number_ring}) <= n",
```
```   552       "(m::'a::{ordered_idom,number_ring}) = n"] (K Lin_Arith.lin_arith_simproc);
```
```   553
```
```   554 end;
```
```   555
```
```   556 Addsimprocs [Int_Arith.fast_int_arith_simproc];
```